Note that we choose \(\psi \)  without knowing the equation for \(y(x)\) . The function \(y(x)\)  is an observable function, as it is a measured function of the state of the system given in Eq. (REF ). Likewise, as \(\psi \)  is a function of an observable, \(\psi \)  is a function of \(x\)  as well. Therefore, in a data-driven setting, dictionary functions of \(y(x)\)  are also observables of some Koopman generator. Typically, we choose \(N\)  such that \(N>m\) , and so \(\psi (y)\)  is a “lifting” of our data (see the use of “lifting” in [1]}).
